The correct interpretation is that these “negative energy solutions” actually describe the anti-
particle of the electron, i.e. the positron. Its mass is exactly equal to that of the electron, and its
charge is exactly opposite. When the positron was discovered experimentally in the 1930s, Dirac’s
theory was immediately propelled to great fame.
We are now ready to writing the general solution of the free Dirac equation,ψ(x) =∫ d (^3) k
(2π)^3
f(k^0 )
∑
s=1, 2
{
us(k)bs(k)e−ik·x+vs(k)d†s(k)e+ik·x
}
ψ†(x) =
∫ d (^3) k
(2π)^3
f(k^0 )
∑
s=1, 2
{
u†s(k)b†s(k)e+ik·x+v†s(k)ds(k)e−ik·x
}
(23.8)
wherekμ= (k^0 ,~k) andk^0 = +
√
~k (^2) +m (^2) , and exhibits only positive energy solutions. Here we
have included a normalization factorf(k^0 ) whose precise form will be fixed by standard convention
later on.
Given the above expressions for the fieldsψandψ†, we are in a position to evaluate physical
observables, such as the electric chargeQand the HamiltonianH for the free Dirac field. We
shall begin by working out the charge. Substitutingψandψ†given in (23.8) into the definition of
electric charge in (23.1), we find,
Q =
∫
d^3 x∫ d (^3) k
(2π)^3
∫ d (^3) k′
(2π)^3
f(k^0 )f(k
′ 0
)∗
∑
s,s′
{
u†s′(k′)b†s′(k′)e+ik
′·x
+vs†′(k′)ds′(k′)e−ik
′·x}
×
{
us(k)bs(k)e−ik·x+vs(k)d†s(k)e+ik·x}
(23.9)Carrying out the integration overxsets~k′= +~kin the direct terms, and~k′ =−~kin the cross
terms. As a result, we have
Q =
∫
d^3 k
(2π)^3|f(k^0 )|^2∑s,s′{
u†s′(k)us(k)b†s′(k)bs(k) +v†s′(k)vs(k)ds′(k)d†s(k)+u†s′(−k)vs(k)b†s′(−k)d†s(k) +v†s′(−k)us(k)ds′(−k)bs(k)}
(23.10)To simplify this expression further, we need a number of spinor identities, which we shall first
develop.
23.2 Spinor Identities
The staring point is the defining relations for the spinorsusandvs,
(ikμγμ+m)us(k) = 0
(ikμγμ−m)vs(k) = 0 (23.11)